Align subscripts vertically

The mathtools package provides a trick for this:

\documentclass{article}
\usepackage{mathtools} % loads also amsmath
\begin{document}
\[
\adjustlimits\sup_{f\in\mathcal{H}}\lim_{\epsilon\downarrow 0}\sup_{g\in\mathcal{H},0<\|g\|<\epsilon} \text{something}
\]
\end{document}

The \adjustlimits command should precede a pair of operators where you want the subscripts to be aligned.

Alternatively, you can define a new operator to be used in cases like these:

\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator*{\plim}{lim\vphantom{p}}

\begin{document}
\[
\sup_{f\in\mathcal{H}}\plim_{\epsilon\downarrow 0}\sup_{g\in\mathcal{H}, 0<\|g\|<\epsilon} \text{something}
\]
\end{document}

About avoiding $$ in LaTeX, please see Why is \[ … \] preferable to $$?

Also it's preferable to write \mathcal{H} with braces, to make clearer what symbol is involved.

I set \arraystretch outside of the environment, so it will affect everything, but I understand it can also be set inside the environment for local effect.

\documentclass{article}
\usepackage{scalerel}
\usepackage{amsmath}
\begin{document}    

Was:
$$\sup_{f\in\mathcal H}\lim_{\epsilon\downarrow 0}\sup_{g\in\mathcal
H,0<\|g\|<\epsilon} \text{something}$$

Is:
\renewcommand\arraystretch{0.6}
\setlength\arraycolsep{0.3ex}
\[
\begin{array}{cccl}
\sup & \lim & \sup & \text{something}\\
\scriptstyle f\in\mathcal H & \scriptstyle \epsilon\downarrow 0 & 
\scriptstyle g\in\mathcal H, 0<\|g\|<\epsilon & \\
\end{array}
\]

\end{document}